12 Time-Independent Approximation Methods 375 12.1 Perturbation Theory 375 Nondegenerate Perturbation Theory 12.1.2 Degenerate Perturbation Theory 12.2 The Variational Method Problems 13 Applications of Time-Independent Approximation Methods 13.1 Hydrogen Atoms Breaking the Degeneracy--Fine Structure 13.2 Spin-Orbit Coupling and the Shell Model of the Nucleus 13. 3 Helium Atoms 411 The Ground State 411 3.3.2 Excited States 13. 4 Multielectron Atoms 422 13.5 Retrospective 13.6 References Problems 14 Atoms in external fields 14.1 Hydrogen Atoms in External Fields 431 4.1.1 Electric fields the Stark effect 431 2 Magnetic Fields-The Zeeman Effect 14.2 Multielectron Atoms in External Magnetic Fields 442 14.3 Retrospective 46 14.4 References 446 Problems 15 Time-Dependent perturbations 449 15.1 Time Dependence of the State Vector 15.2 Two-State Systems 452 Harmonic Perturbation--Rotating Wave pproximation 452 15.2.2 Constant Perturbation Turned On att=o 455 15.3 Time-Dependent Perturbation Theory 457 15.4 Two-state Systems Using Perturbation Theory Harmonic perturbation 15.4.2 Constant Perturbation Turned On att=0 15.5 Extension to Multistate Systems 464 15.5.1 Harmonic perturbation 15.5.2 Constant perturbation Turned On att=o 465 15.5.3 ransitions to a Continuum of states-The Golden rule 15.6 Interactions of Atoms with Radiation 468 15.6.1 The Nature of Electromagnetic transitionsContents xv 12 Time-Independent Approximation Methods . . . . . . . . . . . . . . . . . . . . . . 375 12.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 12.1.1 Nondegenerate Perturbation Theory . . . . . . . . . . . . . 375 12.1.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . 382 12.2 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 13 Applications of Time-Independent Approximation Methods . . . . . . . . 397 13.1 Hydrogen Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 13.1.1 Breaking the Degeneracy—Fine Structure . . . . . . . . 397 13.2 Spin–Orbit Coupling and the Shell Model of the Nucleus . . . . . . 409 13.3 Helium Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 13.3.1 The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 13.3.2 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 13.4 Multielectron Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 13.5 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 13.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 14 Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 14.1 Hydrogen Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . . . 431 14.1.1 Electric Fields—the Stark Effect . . . . . . . . . . . . . . . . 431 14.1.2 Magnetic Fields—The Zeeman Effect . . . . . . . . . . . 436 14.2 Multielectron Atoms in External Magnetic Fields . . . . . . . . . . . . 442 14.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 14.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 15 Time-Dependent Perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 15.1 Time Dependence of the State Vector . . . . . . . . . . . . . . . . . . . . . . . 449 15.2 Two-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 15.2.1 Harmonic Perturbation—Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 15.2.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 455 15.3 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . 457 15.4 Two-state Systems Using Perturbation Theory . . . . . . . . . . . . . . . 459 15.4.1 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 459 15.4.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 462 15.5 Extension to Multistate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 15.5.1 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 464 15.5.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 465 15.5.3 Transitions to a Continuum of States—The Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 15.6 Interactions of Atoms with Radiation . . . . . . . . . . . . . . . . . . . . . . . 468 15.6.1 The Nature of Electromagnetic Transitions . . . . . . . 469